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©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter Calibration An introductory study of measurement uncertainty and its application to digital multimeter calibration Teleconference: US & Canada Toll Free Dial-In Number: 1-(866) 230-5936 International Dial-In Number:+1-281-913-1100 Conference Code: 1010759559
Transcript
Page 1: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

Teleconference

US amp Canada Toll Free Dial-In Number 1-(866) 230-5936

International Dial-In Number+1-281-913-1100

Conference Code 1010759559

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 2

Welcome

Greetings from ndash

Fluke Corporation

Everett Washington USA

We are very pleased to bring you this

presentation on measurement

uncertainty for DMM Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3

Welcome

This presentation is based on Flukersquos

extensive experience with

minus Use and design of calibration

Instruments

minus Our experience and understanding of the

problems faced when applying

measurement uncertainty for both

regular and accredited metrology

Thanks for your time we hope you find it

both valuable and useful

Welcome and Thanks

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4

Presented by

Flukersquos Calibration Business Unit

and Jack Somppi Electrical Calibration Instruments

Product Line Manager

jacksomppiflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

bull Choice of Audio ndash VOIP or Teleconference

minus VOIP receives audio only while teleconference is two way

sound

bull Donrsquot mute your phone if you have background

music enabled

bull Use QampA or chat to send me questions or request

clarification

bull There will be an opportunity throughout the

discussion to pause and ask questions

bull You can view the material using either full screen

or multi window methods

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 2: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 2

Welcome

Greetings from ndash

Fluke Corporation

Everett Washington USA

We are very pleased to bring you this

presentation on measurement

uncertainty for DMM Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3

Welcome

This presentation is based on Flukersquos

extensive experience with

minus Use and design of calibration

Instruments

minus Our experience and understanding of the

problems faced when applying

measurement uncertainty for both

regular and accredited metrology

Thanks for your time we hope you find it

both valuable and useful

Welcome and Thanks

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4

Presented by

Flukersquos Calibration Business Unit

and Jack Somppi Electrical Calibration Instruments

Product Line Manager

jacksomppiflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

bull Choice of Audio ndash VOIP or Teleconference

minus VOIP receives audio only while teleconference is two way

sound

bull Donrsquot mute your phone if you have background

music enabled

bull Use QampA or chat to send me questions or request

clarification

bull There will be an opportunity throughout the

discussion to pause and ask questions

bull You can view the material using either full screen

or multi window methods

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 3: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 3

Welcome

This presentation is based on Flukersquos

extensive experience with

minus Use and design of calibration

Instruments

minus Our experience and understanding of the

problems faced when applying

measurement uncertainty for both

regular and accredited metrology

Thanks for your time we hope you find it

both valuable and useful

Welcome and Thanks

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4

Presented by

Flukersquos Calibration Business Unit

and Jack Somppi Electrical Calibration Instruments

Product Line Manager

jacksomppiflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

bull Choice of Audio ndash VOIP or Teleconference

minus VOIP receives audio only while teleconference is two way

sound

bull Donrsquot mute your phone if you have background

music enabled

bull Use QampA or chat to send me questions or request

clarification

bull There will be an opportunity throughout the

discussion to pause and ask questions

bull You can view the material using either full screen

or multi window methods

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 4: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 4

Presented by

Flukersquos Calibration Business Unit

and Jack Somppi Electrical Calibration Instruments

Product Line Manager

jacksomppiflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

bull Choice of Audio ndash VOIP or Teleconference

minus VOIP receives audio only while teleconference is two way

sound

bull Donrsquot mute your phone if you have background

music enabled

bull Use QampA or chat to send me questions or request

clarification

bull There will be an opportunity throughout the

discussion to pause and ask questions

bull You can view the material using either full screen

or multi window methods

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 5: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 5

Web seminar etiquette

bull Choice of Audio ndash VOIP or Teleconference

minus VOIP receives audio only while teleconference is two way

sound

bull Donrsquot mute your phone if you have background

music enabled

bull Use QampA or chat to send me questions or request

clarification

bull There will be an opportunity throughout the

discussion to pause and ask questions

bull You can view the material using either full screen

or multi window methods

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 6: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 6

Applying Measurement Uncertainty To Digital Multimeter Calibration

An introductory study of measurement

uncertainty and its application to digital

multimeter calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 7: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 7

Objectives

In this session you will -

bull Be introduced to the concept of measurement

uncertainty and why it is important

bull Observe the basic elements that influence

measurement uncertainty for DMM calibration

applications

bull Study a simple but detailed example of calculating

measurement uncertainty

bull Consider some benefits of automating measurement

uncertainty calculations

bull Receive a variety of references for further research on

this topic

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 8: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 8

Benefits

bull Introduce measurement uncertainty to

those in calibrationmetrology who are

not familiar with it

bull Understand why measurement

uncertainty is important for quality

metrology

bull Understand measurement uncertainty

with respect to DMM calibration

bull Appreciate to the benefits of automation

bull Have technical references for more

detailed information

bull Obtain copies of this presentation via

email

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 9: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 9

Measurement Uncertainty amp Why It Is Important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 10: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 10

Facts regarding measurement -

bull Can you ever measure the true value of

something

minus No there will always be errors

bull How important is this fact

minus Very important as measurement is never complete

unless you know how good it is

bull How is this taken into account in todayrsquos

calibration amp metrology

minus By applying amp documenting the measurement uncertainty

process to the tests being done

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 11: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 11

Measurement uncertainty in metrology todayhellip

Measurement errors were not rigorously evaluated in all cases Often in industrial labs accuracy ratio analysis (referred to as TURrsquos or TARrsquos or TSRrsquos) had been frequently used to evaluate the significance of the calibratorrsquos errors on the measurements Other errors were sometimes ignored

Individually analyzed calculated amp documented measurement uncertainties are more thorough and are required to be considered - as stated in

minus ANSIISOIEC 170252005 General Requirements for the Competence of Testing and Calibration Laboratories

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 12: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 12

ISO 17025 ndash about measurement uncertaintyhellip

546 Estimation of uncertainty of

measurement

minus 5461 A calibration laboratory or a testing

laboratory performing its own calibrations shall

have and shall apply a procedure to estimate the

uncertainty of measurement for all calibrations

and types of calibrations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 13: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 13

hellip about the sources of uncertaintyhellip

ISO 17025 Section 5463

minus NOTE 1 Sources contributing to the uncertainty

include but are not necessarily limited to

bull The reference standards and reference

materials used

bull Methods and equipment used

bull Environmental conditions

bull Properties and condition of the item being

tested or calibrated

bull Operator

There are many contributors to uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 14: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 14

ISO 17025 Section 5104

Calibration Certificates shall include hellip

for the interpretation of calibration results

a The conditions of the test

b The uncertainty of measurement amp

compliance statements to metrological standards

c Evidence of traceability

When statements of compliance are made the

uncertainty of measurement shall be taken into account

hellipabout calibration certificateshellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 15: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 15

An example of an accredited calibration certificate ndash

ldquoMeasurement uncertainties at the

time of test are given in the following

pages where applicable They are

calculated in accordance with the

method described in NIST TN1297

for a confidence level of 95 using a

coverage factor of approximately 2

(K=2)rdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 16: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 16

To summarize the importance of measurement uncertaintyhellip

From the NPL UK - ldquoA Beginners Guide to

Uncertainty of Measurementrdquo

bull Uncertainty of a measurement tells us something about

its quality

bull Uncertainty of measurement is the doubt that exists

about the results of any measurement

bull For every measurement ndash even the most careful ndash there

is always a margin of doubt

bull You need to know the uncertainty before you can

decide whether the tolerance is met

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 17: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 17

ldquoHow is this Measurement Uncertainty obtainedrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 18: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 18

Properly Calculating Measurement Uncertainty ndash a topic often discussed amp

debated among metrologists

Initially there were no standardized

process to quantify measurement

uncertaintyhellip

But a standard technique was agreed

upon amp published in October 1993

ISO Guide 98 - Guide to the

Expression of Uncertainty in

Measurement (aka GUM)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 19: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 19

In the USA refer to one of the Guides

relating to expressing of Uncertainty in

Measurement

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Recommendation Refer to the GUMs -

Internationally many metrology

organizations publish similar GUMs

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 20: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 20

Questions - about measurement uncertainty or why it is important

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 21: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 21

Measurement Uncertainty amp Calibrating DMMs

A study of applying the GUM to DMM

calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 22: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 22

First ndash lets look at the concept

Our initial look ndash

bull Consider verifying a

precision digital multimeter

bull With a hypothetical study

of verifying the DMMrsquos

measurement performance

at 100 millivolts DC

bull Letrsquos briefly look at what

measurement uncertainty

could be in this case

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 23: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 23

Some sources of measurement ldquodoubtrdquo when verifying a DMM

bull The most obvious amp significant sources of doubt

minus Inaccuracy of the calibratorrsquos output value

bull 1000000 mV might actually be 1000000 mV 0030 mV

minus Repeatability or randomness in measurement values from the DMM

bull 1000003 mV 999995 mV 1000010 mV etc

minus Resolution or sensitivity limits on the DMM

bull Itrsquos value is frac12 the least significant digit

bull in this example it represents 005 V

bull Many other factors that could also contribute to uncertainty

minus ambient temperature effects thermal emfs noise loading power line

conditions etc

bull Consider all factors and include if they significantly contribute to

measurement uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 24: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 24

The GUMs classify two types of measurement uncertainty

bull Type A uncertainty ndash errors that can be statistically evaluated from the set of measurement data (Often considered as random uncertainty)

minus For example Repeatability of the measurement (influenced by dmm

characteristics signal stability jitter noise etc)

bull Type B uncertainties ndash estimates of errors influencing the

measurement that are not directly observed from the

measurement data (Often considered as systematic

uncertainty)

minus Errors of the calibrating standards (performance specifications for

accuracy changes over time and other conditions)

minus Inherent limitations of the unit being tested (DMM resolution

limitations)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 25: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 25

bull To quantify uncertainty the various sources of uncertainty need

to be quantified evaluated amp combined

bull Calculate a combined estimate of all the individual A and B types

of uncertainties

bull This combined uncertainty is

minus a basic estimate (representing one statistical standard deviation)

minus usually the RSS of all individual uncertainties

(Combining uncertainties using such an RSS technique applies to

uncertainties with standard relationships and are independent)

22

3

2

2

2

1 nc uuuuu

Combining all the uncertainties

cu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 26: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 26

The expanded uncertainty

bull As mentioned calculations for uc pertain to plusmn one standard

deviation of measurement uncertainties (covering 68 of the population of measurements)

bull Usually it is desired to express uncertainty for a larger population or condition say 95 or 99

bull Expanding the calculated uncertainty through scaling estimates an

uncertainty that covers this larger population - Um

bull A coverage factor k (often equal to 2) would indicate a 95 confidence

ckuUm

68

95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 27: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 27

Now returning to the hellip statement of uncertainty

bull A measurement is complete only when

accompanied by a statement of the uncertainty of the

estimate For example

VDMM = 1000051mV 00004 mV

bull In this case 00004 mV would be the resulting value

of Um calculated as shown below

ckumV Um00040

22

3

2

2

2

1 nuuuuk

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 28: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 28

That describes the general process ndash are we okay so far

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 29: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 29

Next a different and more detailed examplehellip

Examine the use of a Fluke 5500A to verify a 35 digit

DMM at 10 Amps of Alternating Current at 50 Hz

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 30: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 30

bull Type A uncertainty is determined by the statistical

analysis of a series of observations (measurements)

bull Type A uncertainties includes effects from

minus Variations of multiple repeated readings from the UUT

minus Effects of the system noise

minus Noise and short term variation of the standard

bull Now letrsquos examine the basic statistics hellip

The ldquoArdquo portionhellip

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 31: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 31

Measurement Value

1 1007

2 1002

3 1001

4 1006

5 1004

Average 1004

Measured value the average of a series of measurements

AIavg 0410

bull An average of multiple measurements is

a better estimate of the true value than

any individual value

bull As a rule of thumb taking between 4 amp

10 measurements are sufficient

bull Uncertainty improvements for more than

10 have diminishing results

bull In our example 5 readings are

sufficient Any improved uncertainties

for more readings are not significant

versus required measurement

tolerances (a typical DMM specification

for this example test is ~ plusmn25)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 32: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 32

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

Calculating the uncertainty due to measurement repeatability

bull The uncertainty is statistically

analyzed from the measurement

data series

u1 ndash for a normally distributed

population the best estimate of

uncertainty is the experimental

standard deviation of the mean NOTE In the unusual case where

1 the calibrating standard is extremely accurate amp

stable and

2 the repeated test measurement values are

unchanged (or even with only a plusmn one digit

change)

Then this uncertainty can be considered as non

significant

bull One measurement value would be sufficient

bull The type B resolution uncertainty is adequate

Experimental

Standard

Deviation

Experimental

Standard Deviation

of the Mean

1

)(1

2

n

xxn

i

i

s

n

su 1

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 33: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 33

Measurement Value Deviation from

Average

x1 1007 +003

x2 1002 -002

x3 1001 -003

x4 1006 +002

x5 1004 000

x (Average) 1004

s (Estimated Std Dev) 002549

The estimated standard deviation

11

2

)(

n

i

n

i

xxs 255 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 34: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 34

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

Plus there are some other important characteristics to

consider

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

What are these

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 35: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 35

Statistical terms amp concepts

bull Probability Distribution ldquothe scatter of the valuesrdquo

minus Normal or Gaussian

minus Rectangular or Uniform

minus Triangular U or bi-modal hellip

bull Degrees of Freedom ldquohow manyrdquo

minus A value related to the amount of information that was employed in

making the estimate

minus Usually equals the sample size minus one (n-1) for type A uncertainties

and is often considered infinite ( ) for parameters such as

manufacturer specifications

bull Sensitivity Coefficient ldquohow influentialrdquo

minus Change in measurement response divided by the corresponding change

in stimulus (usually a value of 1 in the case we are considering)

For more information see technical references on statistics

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 36: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 36

u1 ndash estimated standard uncertainty

Calculate the Standard Deviation of the Mean

minus Probability Distribution = Normal

minus Sensitivity Coefficient = 1

minus Degrees of Freedom = 4

mAmAs

nu 411

5255

1

Grouped around a value

Direct influence on response

Based on 5 independent

measurements

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 37: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 37

The rdquoBrdquo type of uncertainties hellip

All the other uncertainties that cannot be determined statistically during

the measurement process such as -

minus Calibrator inaccuracy or error

minus Measurement errors due to limitations of the DMMrsquos resolution

minus lead effects thermal emfs loading etc

bull Estimates here are based on scientific judgment using all relevant

information

bull Numerically these are expressed as one standard deviation

estimates for each different uncertainty

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 38: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 38

u2 - uncertainty due to the calibrator inaccuracy

u2 is the plusmn1 sigma estimate of the calibrator error

bull (estimates a plusmn1 standard deviation coverage of

the errors - for 68 of all possible values)

bull based on the specifications for performance at the

specific test setting

minus Start with the manufacturerrsquos recommended specifications

at the test point

minus Adjust as required for any appropriate factors such as

legal traceability limitations improvements for output

characterizations etc

minus Convert to a plusmn one sigma confidence interval basis

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 39: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 39

Refer to the calibrator specifications

bull For this example assume it is a certified calibrator that is routinely

calibrated every year

bull The absolute uncertainty specifications for 10 Amps 50 Hz

006 of output plus 2000 Amps

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 40: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 40

Calculating u2

bull Step 1 Calculate the maximum instrument error per

manufacturerrsquos specifications at the point of test

5500A ndash 1 year specs 10 A 50 Hz

plusmn(006 of 10 A + 2000 μA)

is calculated to be

plusmn(6 mA + 2 mA) = plusmn8 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 41: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 41

Calculating u2

bull Step 2 Convert the specified error to an error value that covers plusmnone standard deviation (or a plusmn1 sigma confidence interval)

minus If no other information is provided by the manufacturer assume a rectangular distribution

plusmn1σ = plusmnspec (radic3)

minus If manufacturer specifies a different distribution such as a normal distribution then calculate as appropriate

For example with a normal distribution at 99

plusmn1σ = plusmnspec (258)

Normal Probability Distribution

1 2 3-123

Uniform or Rectangular

Probability Distribution

Pro

ba

bili

ty o

f O

ccu

rre

nce

Value of Reading

Full width

Mean or

Average reading

-a +a

plusmnspec

limits

plusmnspec limits

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 42: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 42

Flukersquos 5500A specifications

The manufacturerrsquos specs document that specifications are based on a normally distributed 99 confidence interval

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 43: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 43

Calculating u2

bullThe value of u2 is the plusmn1 sigma calibrator spec

5500A ndash 1 year specs 10 A 50 Hz

This u2 value should be smaller than the published spec

With a spec of plusmn8 mA at 99 confidence

divide by 258 to convert to a plusmn1 sigma spec

u2 = 8 mA 258 mA = 31 mA at plusmn1 std dev

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 44: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 44

Summary of u2 ndash

u2 is the plusmn1 sigma estimate of calibrator

specification uncertainty

minus Probability Distribution = Normal ndash as stated in the

manufacturerrsquos information

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu2 13

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 45: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 45

u3 - uncertainty due to UUT measurement limitations

bull Measurements include error due to resolution limits of the UUT -

considered as one half of the LSD

bull The LSD of resolution for this UUT measuring 10 Amps is 10 mA

1000 1000000

LSD (least significant digit)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 46: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 46

Calculating u3

The formula for u3 is

Calculates the standard uncertainty related to one LSD

With an LSD of 10 mA -

u3 = 29 mA at a plusmn1 std dev

3LSD2

1 3 u

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 47: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 47

Summary of u3 ndash

u3 is the plusmn1 sigma estimate of dmm LSD resolution

uncertainty

minus Probability Distribution = Rectangular

minus Sensitivity Coefficient = 1

minus Degrees of Freedom =

mAu3 92

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 48: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 48

This completes the ldquoBrdquo portionhellip

u2 = 31 mA at plusmn1 standard deviation

u3 = 29 mA at plusmn1 standard deviation

bull There are no other ldquoBrdquo uncertainties which are

significant for this particular test (Note It is often good to identify and document the

other possible uncertainties deemed insignificant)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 49: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 49

Combining all uncertainties hellip

A One Standard Deviation Estimate Of

Combined Uncertainty

Standard

Combined

Uncertainty

22

3

2

2

2

1 ncuuuuu

1216 mA 222 9213411

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 50: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 50

Overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 810-3 1 Normal 258 3110-3

Resolution B u3 510-3 1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 121610-3 52

How do you calculate the overall

effective Degrees of Freedom

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 51: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 51

Welch-Satterthwaite formula

bull is the overall effective

degrees of freedom for the

combined uncertainty (uc)

bull The formula considers each

uncertainty each sensitivity

coefficient and each

uncertaintyrsquos specific value

for degrees of freedom to

calculate

N

i i

ii

c

eff

v

xuc

yuv

1

44

4

)(

)(

veff

veff

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 52: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 52

Welch-Satterthwaite formula in our example case

25)10(291)10(311

4

)10(1141

)10(1216434434434

43

veff

3

3

4

3

4

3

2

2

4

2

4

2

1

1

4

1

4

1

4

)()()(

)(

v

xuc

v

xuc

v

xuc

yuc

effv

Our effective degrees of freedom considering all our uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 53: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 53

cm

kuU

Calculating the expanded uncertainty

Level of

Confidence

(percent)

Coverage

factor

k

6827 1

90 1645

95 1960

9545 20

99 2576

9973 3

k is the coverage factor

bull How confident should you be with your measurement results

(68 95 99)

bull 95 confidence is commonly accepted as appropriate

bull Um expresses the uncertainty expanded from a single standard

deviation of 68 to uncertainty value with a higher confidence

bull For a large population with a normal distribution 95 coverage

is calculated by k with a value of 196

(or sometimes 2 for convenience ndash giving 9545)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 54: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 54

Adjusting k for a smaller set of measurements or samples

bull Adjusting k is done using the

studentsrsquo t distribution table

bull A coverage factor adjustment

is needed because our data

set had a fewer number of

values rather than a larger set

(such as 20 50 or 100)

bull The table lists the proper

coverage factor for populations

with smaller degrees of

freedom

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

For our example with the effective degrees of freedom (Veff) of 52

a coverage factor of 257 expands uc to a value with 95 confidence

(compared to 196 for an infinite set of measurementssamples)

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 55: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 55

cm

kuU

Expanded measurement uncertainty calculation

572Um1216 mA

U m 3126 mA

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 56: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 56

Our overall uncertainty budget

Source of

Uncertainty Type Ui

Uncertainty

Value

(Amps)

Sensitivity

Coefficient

Probability

Distribution

Coverage

Factor

Standard

Uncertainty

(Amps)

Degrees

of

Freedom

Repeatability A u1 11410-3 1 Normal 1 11410-3 4

Calibrator B u2 710-3 1 Normal 258 2710-3

Resolution B u3 510-3

1 Rectangular 2910-3

Current

Measurement Combined uC - -

Assumed

Normal - 12110-3 52

Current

Measurement Expanded Um

312610-3 - Assumed

Normal 257 - 52

3

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 57: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 57

mavg UII

Final results -

bull The final measurement value including the

measurement uncertainty from the series of DMM

measurements of the calibrator

AmpsI 0410 0031 At a level of confidence of 95

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 58: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 58

What if more measurements were taken does that improve the uncertainty

Increased degrees of freedom

Veff = 5 10 20 or 100

Causes marginal improvements

in k and in

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull 9 measurements Veff = 103

minus k = 223 = 27 mA (4 mA better)

bull 17 measurements Veff = 207

minus k = 209 = 25 mA (2 mA better)

bull 78 measurements Veff = 1009

minus k = 1984 = 24 mA (1 mA better)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

U m

U m

U m

So improves only 7 mA by taking

73 more measurements U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 59: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 59

Does improving beyond plusmn31 mA by taking more measurements have any practical value

Whatrsquos the value of increasing

Veff from 5 to

The test tolerance is plusmn250 mA

bull 5 measurements Veff = 52

minus k = 257 = 31 mA

bull With a = 31mA

the test ratio is already 81

(TUR = Test Spec divide Total Uncertainty

025A divide 31mA = 806)

Fraction p in percentDegrees of

freedom 6827 90 95 9545 99 9973

1 184 631 1271 1397 6366 2358

2 132 292 43 453 992 1921

3 12 235 318 331 584 922

4 114 213 278 287 46 662

5 111 202 257 265 403 551

6 109 194 245 252 371 49

7 108 189 236 243 35 453

8 107 186 231 237 336 428

9 106 183 226 232 325 409

10 105 181 223 228 317 396

20 103 172 209 213 285 342

50 101 168 201 205 268 316

100 1005 166 1984 2025 2626 3077

1 1645 196 2 2576 3

U m

U m

AmpsI 0410 0031

So to satisfy a minimum test ratio of 41

5 measurements are more than adequate

U m

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 60: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 60

Questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 61: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 61

Making The Calculation Of Measurement Uncertainty Simpler

What can you do to automate this

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 62: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 62

Automation alternatives

bull A custom program

designed for a specific

requirement

bull A custom spreadsheet for

analysis

bull A commercial metrology

based software package

such as

Flukersquos METCAL Plus

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 63: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 63

METCAL automates the uncertainty calculations

Post test summary of

10000A 50Hz

Including

5 reading average

Calculated combined

standard uncertainty

How does this work

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 64: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 64

METCAL manages amp analyses the uncertainties

Number of Measurements = 5

Value 1 = 1007

Value 2 = 1001

Value 3 = 1002

Value 4 = 1004

Value 5 = 1006

UUT Indicated = 1004

Standard Deviation = 002549509757

Standard uncertainty = 001140175425

Sensitivity Coefficient = 1

Degrees of Freedom = 4

System Actual = 10

System Accuracy = 0008

Confidence interval of spec = 258

1 Sigma Spec = 0003126379456

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

UUT Resolution = 001

Resol Standard Uncertainty = 0002886751346

Sensitivity Coefficient = 1

Degrees of Freedom = 1e+200

Combined Std Uncertainty = 001216490061

Effective Deg of Freedom = 5186506

Standard Uncertainty = 001207040471

Coverage Factor = 2567104753

Expanded Uncertainty = 0031263794

Calculated

Total

Uncertainty

Repeatability

Uncertainty

Calibrator

Uncertainty

Resolution

Uncertainty

Measurement

Details

With METCAL the user configures

bull Specific statistics used

bull Confidence Coverage

bull Number of measurements

bull Accuracy of the standard

In the cal or test procedure you also specify test parameters

bull Test point

bull UUT resolution

In the test process METCAL provides the uncertainty details (our example is shown to the right)

Details are permanently stored in the data base They accessible for reports amp future analysis

METCAL Data for

our example

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 65: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 65

ldquoAutomationrdquo ndash some words of wisdom

bull Remember it is always the metrologistrsquos responsibility to

insure proper calculation of measurement uncertainty

minus Every lab has unique characteristics which must be supported

minus Configuring the measurement characteristics is also unique

minus Defining the specific error budget for the test

minus Configuring the specific measurement uncertainty parameters

bull There should be definite information to support answering

any auditorrsquos questions

bull Keep records of the procedurersquos measurement design with

an uncertainty error budget

bull Be able to demonstrate the reasonableness of the testrsquos

uncertainties

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 66: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 66

Benefits of METCAL automation

bull Automation simplifies a structured calculation process

bull Usable for manual semi automated or fully automated testing methods

bull METCAL provides flexibility to customize the calculation process amp factors

bull METCALrsquos database stores all the information for future reference

bull Report writing flexibility permits properly configured certificates and data summaries

bull Lets the technical staff concentrate on the test quality rather than the rote mathematical amp statistical processes

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 67: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 67

Automation questions

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 68: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 68

Conclusion amp Review ndash What have we done

bull Topics

minus Measurement uncertainty amp why it is important

minus How measurement uncertainty obtained

minus Examples on measurement uncertainty amp calibrating DMMs

minus Benefits of automating

bull Measurement Uncertainty is becoming an essential consideration in all metrology amp calibration measurements

bull Measurement results are considered incomplete without a quoted uncertainty

bull Calculations usually require a statistical process on multiple measurements for each test

bull Automation can be a valuable support for measurement uncertainty calculations

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 69: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 69

Obtain a copy of the GUMs amp

other references for details

ANSINCSL Z5402-1997 (R2002) US

Guide to Expression of Uncertainty in

Measurement httpwwwncsliorg and find it in the store

under NCSLI publications

NIST Technical Note 1297 httpwwwphysicsnistgovPubsguidelines

contentshtml

Where to go from here

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 70: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 70

For more information (1) -

bull Chapters 20-22 on Statistics amp

Uncertainty in the text book

Calibration Philosophy in

Practice 2nd Edition

bull Flukersquos Training Course ndash Cal Lab

Management for the 21st Century

bull Various reference material under

technical papers at the resource

library on Flukersquos web site

httpwwwflukecom

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 71: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 71

For more information (2) -

bull EA-402 ldquoExpression of the Uncertainty of Measurement of Calibrationrdquo httpwwweuropean-accreditationorg

bull UKAS Publication LAB-12 ldquoThe Expression of Uncertainty In Testingrdquo httpwwwukascom

bull NPL UK - ldquoA Beginners Guide to Uncertainty of Measurementrdquo httpwwwnplcouknpl

bull Flukersquos ldquoCalibration ndash Philosophy in Practice Second Editionrdquo

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 72: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 72

Still more references (3)

bull NCSL International RP-12 - Determining amp

Reporting Measurement Uncertainties httpswwwncsliorg

bull NIST Website Essentials of expressing

measurement uncertainty

httpphysicsnistgovcuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 73: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 73

Questions

22

3

2

2

2

1 nc uuuuu

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 74: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 74

Fluke Calibration Web Seminar Series

For information amp reservations to attend our

seminars go to wwwflukecalcom click

on the menu selection ldquoEvents amp

Trainingrdquo and click on the ldquoWeb

Seminarsrdquo selection and again click on

the desired seminar selection

Our Seminar Topics Include

bull Precision Measurement Techniques

bull Oscilloscope Calibration

bull General Metrology

bull Temperature Calibration

bull Metrology Software

bull RF Calibration

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 75: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 75

Calibration and metrology training

bull Instructor-Led Classroom Training minus MET-101 Basic Hands-on Metrology (new in 2007)

minus MET-301 Advanced Hands-on Metrology (new in 2007)

minus MET-302 Hands-on Metrology Statistics (new in 2009)

minus Cal Lab Management for the 21st Century

minus Metrology for Cal Lab Personnel (A CCT prep course)

minus METCAL Database and Reports

minus METCAL Procedure Writing

minus METCAL Advanced Programming Techniques

minus On-Site Training

minus Product Specific Training

bull Instructor-Led Web-Based Training minus METCAL Database Web-Based Training

minus METCAL Procedure Development Web-Based Training

bull Self-Paced Web-Based Training minus Introduction to Measurement and Calibration

minus Precision Electrical Measurement

minus Measurement Uncertainty

minus ACDC Calibration and Metrology

minus Metrology for Cal Lab Personnel (A CCT prep course)

bull Self-Paced Training Tools minus METCAL-CBT7 Computer Based Training

minus METCAL-CBTPW Computer-Based Training (new in 2007)

minus Cal-Book Philosophy in Practice textbook More information

wwwflukecalcomtraining

Members of the METSUPPORT Gold and Priority Gold CarePlan support programs receive a 20

discount off any Fluke calibration training course

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom

Page 76: Applying Measurement Uncertainty To Digital …©Fluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 1 Applying Measurement Uncertainty To Digital Multimeter

copyFluke Calibration 2011 Basics Of Measurement Uncertainty for DMM Calibration 76

THANK YOU

For material related to this session visit our web site

httpwwwflukecom

For any questions or copies of this presentation

email inquiries to calibrationseminarsflukecom


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